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© 2017 W.H. Freeman and Company The correct rule for proportions is that we can create confidence intervals for p based on a. if n is at least 10. b. at least 15 successes and 15 failures. c. if n ≥ 30. 8.1 Inference for a Single Proportion The correct rule for proportions is that we can create confidence intervals for p based on a. if n is at least 10. b. at least 15 successes and 15 failures. (correct) c. if n ≥ 30. 8.1 Inference for a Single Proportion In a sample of 20 items, I found six to be defective. In constructing a confidence interval for the proportion of defectives, I should use a. the plus four method. b. the large sample interval. c. neither method. 8.1 Inference for a Single Proportion In a sample of 20 items, I found six to be defective. In constructing a confidence interval for the proportion of defectives, I should use a. the plus four method. (correct) b. the large sample interval. c. neither method. The sample size is larger than 10, but fewer than 10 defectives, so the plus four method will be best. 8.1 Inference for a Single Proportion A sample of 75 students found that 55 of them had cell phones. The margin of error for a 95% confidence interval estimate for the proportion of all students with cell phones is a. 0.084. b. (0.633, 0.833). c. 0.100. 8.1 Inference for a Single Proportion A sample of 75 students found that 55 of them had cell phones. The margin of error for a 95% confidence interval estimate for the proportion of all students with cell phones is a. 0.084. b. (0.633, 0.833). z* 55 20 ˆˆ pq 1.96 75 75 0.100 n 75 c. 0.100. (correct) 8.1 Inference for a Single Proportion Suppose a sample of 155 students at a university were asked their grade in statistics. 21 said they made A’s. Is the use of the Normal approximation justified in this case? a. Yes, because successes and failures are both more than 10. b. Yes, because total sample size is greater than 30. c. No, because number of successes is less than 30. 8.1 Inference for a Single Proportion Suppose a sample of 155 students at a university were asked their grade in statistics. 21 said they made A’s. Is the use of the Normal approximation justified in this case? a. Yes, because successes and failures are both more than 10. (correct) b. Yes, because total sample size is greater than 30. c. No, because number of successes is less than 30. 8.1 Inference for a Single Proportion It is thought that 12% of all students taking a particular course received a grade of A. In a sample of 155 students, it is found that 21 made an A. What is the test statistic for testing the true proportion is 12%? a. 0.53 b. 0.01 c. 0.57 8.1 Inference for a Single Proportion It is thought that 12% of all students taking a particular course received a grade of A. In a sample of 155 students, it is found that 21 made an A. What is the test statistic for testing the true proportion is 12%? a. 0.53 b. 0.01 0.135 0.12 z 0.57 0.12(1 0.12) 155 c. 0.57 (correct) 8.1 Inference for a Single Proportion You buy a package of 122 Smarties and 19 of them are red. What is a 95% confidence interval for the true proportion of red Smarties? a. (0.092, 0.220) b. (0.103, 0.230) c. (0.085, 0.199) 8.1 Inference for a Single Proportion You buy a package of 122 Smarties and 19 of them are red. What is a 95% confidence interval for the true proportion of red Smarties? a. (0.092, 0.220) (correct) b. (0.103, 0.230) c. (0.085, 0.199) pˆ (1 pˆ ) pˆ z n pˆ 19 /122 * 8.1 Inference for a Single Proportion We want to construct a 95% confidence interval for the true proportion of all adult males who have spent time in prison, with a margin of error of 0.02. From previous studies, we believe the proportion to be somewhere around 0.07. The required sample size is, therefore, a. 620. b. 626. c. 632. 8.1 Inference for a Single Proportion We want to construct a 95% confidence interval for the true proportion of all adult males who have spent time in prison, with a margin of error of 0.02. From previous studies, we believe the proportion to be somewhere around 0.07. The required sample size is, therefore, a. 620. b. 626. (correct) c. 632. * z 2 * n ( ) ( p )(1 p* ) m n 9604(0.07)(0.93) 8.1 Inference for a Single Proportion In a sample of 446 students, 246 ate breakfast. Can we can conclude that more than 50% of all students eat breakfast? We will test H0: p = 0.50 Ha: p > 0.50. What is the value of the test statistic? a. 2.11 b. 5.62 c. 2.31 8.1 Inference for a Single Proportion In a sample of 446 students, 246 ate breakfast. Can we can conclude that more than 50% of all students eat breakfast? We will test H0: p = 0.50 Ha: p > 0.50. What is the value of the test statistic? a. 2.11 (correct) b. 5.62 c. 2.31 0.55 0.50 z 2.11 0.50(0.50) 446 8.1 Inference for a Single Proportion In a sample of 446 students, 246 ate breakfast. Can we can conclude that more than 50% of all students eat breakfast? We will test H0: p = 0.50 Ha: p > 0.50. What is the p-value of the test? a. 0.0174 b. 0.1765 c. 0.2876 8.1 Inference for a Single Proportion In a sample of 446 students, 246 ate breakfast. Can we can conclude that more than 50% of all students eat breakfast? We will test H0: p = 0.50 Ha: p > 0.50. What is the p-value of the test? a. 0.0174 (correct) b. 0.1765 p ( z 2.11) 1 0.9826 0.0174 c. 0.2876 8.1 Inference for a Single Proportion An article about a new drug stated that “the incidence of side effects was similar to placebo, P-value > 0.05.” With the information given: a. one should reject the null hypothesis of no difference at 10%. b. one should not reject the null hypothesis of no difference at 10%. c. There is not enough information given. 8.1 Inference for a Single Proportion An article about a new drug stated that “the incidence of side effects was similar to placebo, P-value > 0.05.” With the information given: a. one should reject the null hypothesis of no difference at 10%. b. one should not reject the null hypothesis of no difference at 10%. c. There is not enough information given. (correct) We know the P-value is more than 5%, but we do not know how much more. Without the actual P-value for the test, we cannot make a determination. 8.1 Inference for a Single Proportion A sample of 42 parts from an assembly line are checked and four are found to be defective. Find a 90% confidence interval for the true proportion of defectives. a. (0.021, 0.170) b. (0.019, 0.155) c. (0.049, 0.212) 8.1 Inference for a Single Proportion A sample of 42 parts from an assembly line are checked and four are found to be defective. Find a 90% confidence interval for the true proportion of defectives. a. (0.021, 0.170) Use the plus four estimate … X 2 p n4 b. (0.019, 0.155) c. (0.049, 0.212) (correct) 8.1 Inference for a Single Proportion Drug-sniffing dogs must be 95% accurate. A new dog is being tested and is right in 46 of 50 trials. Find a 95% confidence interval for the proportion of times the dog will be correct. a. (0.845, 0.995) b. (0.805, 0.973) c. (0.819, 0.959) 8.1 Inference for a Single Proportion Drug-sniffing dogs must be 95% accurate. A new dog is being tested and is right in 46 of 50 trials. Find a 95% confidence interval for the proportion of times the dog will be correct. a. (0.845, 0.995) Use the plus four estimate … p X 2 n4 b. (0.805, 0.973) (correct) c. (0.819, 0.959) 8.1 Inference for a Single Proportion C Drug-sniffing dogs must be 95% accurate. A new dog is being tested and is right in 46 of 50 trials. Find a 95% confidence interval for the proportion of times the dog will be correct. a. (0.845, 0.995) b. (0.805, 0.973) c. (0.819, 0.959) 8.1 Inference for a Single Proportion C Drug-sniffing dogs must be 95% accurate. A new dog is being tested and is right in 46 of 50 trials. Find a 95% confidence interval for the proportion of times the dog will be correct. Use the plus four estimate … p X 2 n4 a. (0.845, 0.995) b. (0.805, 0.973) (correct) c. (0.819, 0.959) 8.1 Inference for a Single Proportion A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that he had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. The standard error for a confidence interval for the candidate’s latest support level is a. 0.016. b. 0.020. c. 0.025. 8.1 Inference for a Single Proportion A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that he had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. The standard error for a confidence interval for the candidate’s latest support level is a. 0.016. (correct) b. 0.020. c. 0.025. SE ( pˆ ) pˆ (1 pˆ ) n 8.1 Inference for a Single Proportion C A noted psychic was tested for ESP. The psychic was presented with 400 cards face down and asked to determine if each card was marked with one of four different symbols. The psychic was correct in 120 cases. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Suppose you wish to see if there is evidence that the psychic was doing better than just guessing. To do this, you test the hypotheses: H0: p = 0.25, Ha: p > 0.25 The P-value of your test is a. 0.0104. b. 0.0146. c. 0.9896. 8.1 Inference for a Single Proportion C A noted psychic was tested for ESP. The psychic was presented with 400 cards face down and asked to determine if each card was marked with one of four different symbols. The psychic was correct in 120 cases. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Suppose you wish to see if there is evidence that the psychic was doing better than just guessing. To do this, you test the hypotheses: H0: p = 0.25, Ha: p > 0.25 The P-value of your test is a. 0.0104. (correct) b. 0.0146. c. 0.9896. 8.1 Inference for a Single Proportion A sample of 425 products off of an assembly line are checked and 21 are found to be defective. The margin of error for a 95% confidence interval for the proportion of defectives is a. 0.021. b. 0.011. c. 0.001. 8.1 Inference for a Single Proportion A sample of 425 products off of an assembly line are checked and 21 are found to be defective. The margin of error for a 95% confidence interval for the proportion of defectives is a. 0.021. (correct) b. 0.011. c. 0.001. pˆ (1 pˆ ) ME z * n 8.1 Inference for a Single Proportion In a random sample of 600 television viewers contacted by phone in a certain suburban area, 210 were watching the movie on Channel 12. Give a 95% confidence interval for the total proportion of viewers watching the movie. a. (.31, .39) b. (.32, .38) c. (.34, .36) 8.1 Inference for a Single Proportion In a random sample of 600 television viewers contacted by phone in a certain suburban area, 210 were watching the movie on Channel 12. Give a 95% confidence interval for the total proportion of viewers watching the movie. a. (.31, .39) (correct) b. (.32, .38) c. (.34, .36) pˆ (1 pˆ ) pˆ z * , z 1.96, pˆ .35 n 8.1 Inference for a Single Proportion Suppose you want to know which of two manufacturing methods will be better. You create 10 prototypes using the first process and 10 using the second. There were three defectives in the first batch and five in the second. A 90% confidence interval for the difference in the two proportions is (-0.493, 0.159). What conclusion should you make about the two manufacturing processes? a. The first method is better. b. The second method is better. c. Both methods may be equivalent. 8.2 Comparing Two Proportions Suppose you want to know which of two manufacturing methods will be better. You create 10 prototypes using the first process and 10 using the second. There were three defectives in the first batch and five in the second. A 90% confidence interval for the difference in the two proportions is (-0.493, 0.159). What conclusion should you make about the two manufacturing processes? a. The first method is better. b. The second method is better. c. Both methods may be equivalent. (correct) 8.2 Comparing Two Proportions You want to know which of two manufacturing methods will be better. You create 10 prototypes using the first process and 10 using the second. There were three defectives in the first batch and five in the second. Find a 95% confidence interval for the difference in the proportion of defectives. a. (-0.62, 0.22) b. (-0.56, 0.22) c. (-0.493, 0.160) 8.2 Comparing Two Proportions You want to know which of two manufacturing methods will be better. You create 10 prototypes using the first process and 10 using the second. There were three defectives in the first batch and five in the second. Find a 95% confidence interval for the difference in the proportion of defectives. a. (-0.62, 0.22) The numbers of trials and defectives are small, so we should use the plus four estimates, where we add two trials to each sample and one defective to each sample. b. (-0.56, 0.22) (correct) c. (-0.493, 0.160) 8.2 Comparing Two Proportions C You want to know which of two manufacturing methods will be better. You create 10 prototypes using the first process, and 10 using the second. There were three defectives in the first batch and five in the second. Find a 95% confidence interval for the difference in the proportion of defectives. a. (-0.62, 0.22) b. (-0.56, 0.22) c. (-0.493, 0.160) 8.2 Comparing Two Proportions C You want to know which of two manufacturing methods will be better. You create 10 prototypes using the first process, and 10 using the second. There were three defectives in the first batch and five in the second. Find a 95% confidence interval for the difference in the proportion of defectives. The numbers of trials and defectives are small, so we should use the plus four estimates, where we add two trials to each sample and one defective to each sample. a. (-0.62, 0.22) b. (-0.56, 0.22) (correct) c. (-0.493, 0.160) 8.2 Comparing Two Proportions A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that he had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. We want to know if his support has decreased. In computing a test of hypothesis with 0 : 1 = 2 , what is the estimate of the overall proportion? a. 52% b. 52.5% c. 51.5% 8.2 Comparing Two Proportions A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that he had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. We want to know if his support has decreased. In computing a test of hypothesis with 0 : 1 = 2 , what is the estimate of the overall proportion? a. 52% b. 52.5% 1 = .54 ∗ 600 = 324 2 = 0.5 ∗ 1030 = 515 So, total of 600 + 1030 = 1630. Hence, 324+515 = 1630 ≈ .515 c. 51.5% (correct) 8.2 Comparing Two Proportions A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that he had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. We want to know if his support has decreased. The test statistic is a. z = 1.56. b. z = -2.57. c. z = -1.55. 8.2 Comparing Two Proportions A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that he had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. We want to know if his support has decreased. The test statistic is a. z = 1.56. (correct) b. z = -2.57. c. z = -1.55. z .54 .50 1 1 .515 (1 .515) 600 1030 pˆ 324 515 .515 1630 8.2 Comparing Two Proportions C A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that he had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. We want to know if his support has decreased. The test statistic is a. z = 1.56. b. z = -2.57. c. z = -1.55. 8.2 Comparing Two Proportions C A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that he had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. We want to know if his support has decreased. The test statistic is a. z = 1.56. (correct) b. z = -2.57. c. z = -1.55. 8.2 Comparing Two Proportions Based on surveys conducted in 1989 and 1999, a researcher compared the proportion of high-school-age females interested in a career in science in 1989 with the proportion in 1999. He concluded that the proportions were not significantly different at the a = 0.05 level because the P-value was 0.121. Assuming the surveys were simple random samples from the appropriate populations, we may conclude a. that the probability of observing a difference at least as large as that observed by the researcher if, in fact, the two proportions were equal is 0.121. b. that in repeated sampling, the researcher would obtain the difference actually observed in approximately 12.1% of the samples. c. very little. Without knowing if the observed difference is practically significant, we cannot assess whether the results are statistically significant. 8.2 Comparing Two Proportions Based on surveys conducted in 1989 and 1999, a researcher compared the proportion of high-school-age females interested in a career in science in 1989 with the proportion in 1999. He concluded that the proportions were not significantly different at the a = 0.05 level because the P-value was 0.121. Assuming the surveys were simple random samples from the appropriate populations, we may conclude a. that the probability of observing a difference at least as large as that observed by the researcher if, in fact, the two proportions were equal is 0.121. (correct) b. that in repeated sampling, the researcher would obtain the difference actually observed in approximately 12.1% of the samples. c. very little. Without knowing if the observed difference is practically significant, we cannot assess whether the results are statistically significant. 8.2 Comparing Two Proportions C An SRS of 100 of a certain popular model car in 1993 found that 20 had a certain minor defect in the brakes. An SRS of 400 of this model car in 1994 found that 50 had the minor defect in the brakes. Let p1 and p2 be the proportion of all cars of this model in 1993 and 1994, respectively, that actually contain the defect. A 90% confidence interval for p1 – p2 is a. 0.075 ± 0.084. b. 0.075 ± 0.071. c. 0.075 ± 0.043. 8.2 Comparing Two Proportions C An SRS of 100 of a certain popular model car in 1993 found that 20 had a certain minor defect in the brakes. An SRS of 400 of this model car in 1994 found that 50 had the minor defect in the brakes. Let p1 and p2 be the proportion of all cars of this model in 1993 and 1994, respectively, that actually contain the defect. A 90% confidence interval for p1 – p2 is a. 0.075 ± 0.084. b. 0.075 ± 0.071. (correct) c. 0.075 ± 0.043. 8.2 Comparing Two Proportions C A manufacturer receives parts from two suppliers. An SRS of 400 parts from supplier 1 finds 20 defective. An SRS of 100 parts from supplier 2 finds 10 defective. Let p1 and p2 be the proportion of all parts from suppliers 1 and 2, respectively, that are defective. Is there evidence of a difference in the proportion of defective parts produced by the two suppliers? To determine this, you test the hypotheses: H 0 : p1 = p2 H a : p1 ≠ p2 The P -value of your test is a. 0.1164. b. 0.060. c. 0.0301. 8.2 Comparing Two Proportions C A manufacturer receives parts from two suppliers. An SRS of 400 parts from supplier 1 finds 20 defective. An SRS of 100 parts from supplier 2 finds 10 defective. Let p1 and p2 be the proportion of all parts from suppliers 1 and 2, respectively, that are defective. Is there evidence of a difference in the proportion of defective parts produced by the two suppliers? To determine this, you test the hypotheses: H 0 : p1 = p2 H a : p1 ≠ p2 The P -value of your test is a. 0.1164. b. 0.060. (correct) c. 0.0301. 8.2 Comparing Two Proportions A psychologist claims she has developed a cognitive-therapy program that is more effective in helping smokers quit smoking than other currently available programs. In particular, she claims that her program is more effective than the nicotine patch, which is widely used by smokers trying to quit. A sample of 75 adult smokers who have indicated a desire to quit is located. The subjects are randomized into two groups. The cognitive-therapy program will be administered to the 38 smokers in the first group, and the 37 smokers in the second group will use the nicotine patch. After a period of 1 year, each subject indicates whether he or she has successfully quit smoking. In the therapy group, 22 people say they have quit smoking, while 17 people who used the patch have quit. What is the value of the test statistic for this claim? a. 1.04 b. 1.14 c. 1.24 8.2 Comparing Two Proportions A psychologist claims she has developed a cognitive-therapy program that is more effective in helping smokers quit smoking than other currently available programs. In particular, she claims that her program is more effective than the nicotine patch, which is widely used by smokers trying to quit. A sample of 75 adult smokers who have indicated a desire to quit is located. The subjects are randomized into two groups. The cognitive-therapy program will be administered to the 38 smokers in the first group, and the 37 smokers in the second group will use the nicotine patch. After a period of 1 year, each subject indicates whether he or she has successfully quit smoking. In the therapy group, 22 people say they have quit smoking, while 17 people who used the patch have quit. What is the value of the test statistic for this claim? a. 1.04 (correct) b. 1.14 c. 1.24 z pˆ1 pˆ 2 0.579 0.459 0.120 1.04 1 1 1 0.013314 1 0.520(0.480) ˆp(1 pˆ ) 38 37 n1 n 2 8.2 Comparing Two Proportions A nutritionist wants to determine if males consume more calcium than females. Of the 180 males in her sample, 99 indicated they consumed the recommended daily intake of calcium, compared with 232 of the 320 females. The nutritionist wants to test whether the proportions of males and females who consume the recommended daily intake of calcium are the same. What null and alternative hypotheses should the nutritionist establish? a. H0: pf = pm vs. Ha: pf > pm b. H0: pf = pm vs. Ha: pf < pm c. H0: pf < pm vs. Ha: pf = pm 8.2 Comparing Two Proportions A nutritionist wants to determine if males consume more calcium than females. Of the 180 males in her sample, 99 indicated they consumed the recommended daily intake of calcium, compared with 232 of the 320 females. The nutritionist wants to test whether the proportions of males and females who consume the recommended daily intake of calcium are the same. What null and alternative hypotheses should the nutritionist establish? a. H0: pf = pm vs. Ha: pf > pm b. H0: pf = pm vs. Ha: pf < pm (correct) c. H0: pf < pm vs. Ha: pf = pm 8.2 Comparing Two Proportions A nutritionist wants to determine if males consume more calcium than females. Of the 180 males in her sample, 99 indicated they consumed the recommended daily intake of calcium, compared with 232 of the 320 females. What is the absolute value of the test statistic? a. 8.27 b. 8.14 c. 3.97 8.2 Comparing Two Proportions A nutritionist wants to determine if males consume more calcium than females. Of the 180 males in her sample, 99 indicated they consumed the recommended daily intake of calcium, compared with 232 of the 320 females. What is the absolute value of the test statistic? a. 8.27 b. 8.14 c. 3.97 (correct) 8.2 Comparing Two Proportions Interest is in determining if a new treatment has a higher success probability than a standard treatment. 150 people are given the new treatment and 100 are given the standard. What is the null and alternative hypotheses? a. H0: pn = ps vs. Ha: pn < ps b. H0: pn = ps vs. Ha: pn > ps c. H0: pn < ps vs. Ha: pn = ps 8.2 Comparing Two Proportions Interest is in determining if a new treatment has a higher success probability than a standard treatment. 150 people are given the new treatment and 100 are given the standard. What is the null and alternative hypotheses? a. H0: pn = ps vs. Ha: pn < ps b. H0: pn = ps vs. Ha: pn > ps (correct) c. H0: pn < ps vs. Ha: pn = ps 8.2 Comparing Two Proportions Interest is in determining if a new treatment has a higher success probability than a standard treatment. 150 people are given the new treatment (115 successes) and 100 are given the standard (65 successes). What is the value of the test statistic? a. 1.41 b. 2.01 c. 1.96 8.2 Comparing Two Proportions Interest is in determining if a new treatment has a higher success probability than a standard treatment. 150 people are given the new treatment (115 successes) and 100 are given the standard (65 successes). What is the value of the test statistic? a. 1.41 b. 2.01 (correct) c. 1.96 8.2 Comparing Two Proportions